The Annals of Probability

When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull?

Abstract

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is related to Gordon’s escape theorem dealing with standard Gaussian matrices. We show that for the random walk $\mathrm{BM}_{n}(i),i\in\mathbb{N}$, the convex hull of the first $C^{n}$ steps (for a sufficiently large universal constant $C$) contains the origin with probability close to one. Moreover, the approach allows us to prove that with high probability the $\pi/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we prove an extension of Gordon’s theorem dealing with a broad class of random matrices; as an application, we show that $C^{n}$ steps are sufficient for the standard walk on $\mathbb{Z}^{n}$ to absorb the origin into its convex hull with a high probability. Finally, we prove that the aforementioned bound is sharp in the following sense: for some universal constant $c>1$, the convex hull of the $n$-dimensional Brownian motion $\operatorname{conv}\{\mathrm{BM}_{n}(t):t\in[1,c^{n}]\}$ does not contain the origin with probability close to one.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 965-1002.

Dates
Revised: November 2015
First available in Project Euclid: 31 March 2017

https://projecteuclid.org/euclid.aop/1490947312

Digital Object Identifier
doi:10.1214/15-AOP1079

Mathematical Reviews number (MathSciNet)
MR3630291

Zentralblatt MATH identifier
1377.52008

Citation

Tikhomirov, Konstantin; Youssef, Pierre. When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull?. Ann. Probab. 45 (2017), no. 2, 965--1002. doi:10.1214/15-AOP1079. https://projecteuclid.org/euclid.aop/1490947312

References

• [1] Candès, E. J. (2014). Mathematics of sparsity (and a few other things). In Proceedings of the International Congress of Mathematicians. Seoul, South Korea.
• [2] Chafaï, D., Guédon, O., Lecué, G. and Pajor, A. (2012). Interactions Between Compressed Sensing Random Matrices and High Dimensional Geometry. Panoramas et Synthèses [Panoramas and Syntheses] 37. Société Mathématique de France, Paris.
• [3] Chandrasekaran, V., Recht, B., Parrilo, P. A. and Willsky, A. S. (2012). The convex geometry of linear inverse problems. Found. Comput. Math. 12 805–849.
• [4] Eldan, R. (2014). Extremal points of high-dimensional random walks and mixing times of a Brownian motion on the sphere. Ann. Inst. Henri Poincaré Probab. Stat. 50 95–110.
• [5] Eldan, R. (2014). Volumetric properties of the convex hull of an $n$-dimensional Brownian motion. Electron. J. Probab. 19 no. 45, 34.
• [6] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I. Third Edition. Wiley, New York.
• [7] Gordon, Y. (1985). Some inequalities for Gaussian processes and applications. Israel J. Math. 50 265–289.
• [8] Gordon, Y. (1988). On Milman’s inequality and random subspaces which escape through a mesh in $\textbf{R}^{n}$. In Geometric Aspects of Functional Analysis (1986/87). Lecture Notes in Math. 1317 84–106. Springer, Berlin.
• [9] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
• [10] Kabluchko, Z. and Zaporozhets, D. (2016). Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls. Trans. Amer. Math. Soc. 368 8873–8899.
• [11] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 23. Springer, Berlin.
• [12] Litvak, A. E., Pajor, A. and Tomczak-Jaegermann, N. (2006). Diameters of sections and coverings of convex bodies. J. Funct. Anal. 231 438–457.
• [13] Matthews, P. (1988). Covering problems for Brownian motion on spheres. Ann. Probab. 16 189–199.
• [14] Mendelson, S. (2014). A remark on the diameter of random sections of convex bodies. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 2116 395–404. Springer, Cham.
• [15] Milman, V. D. (1985). Random subspaces of proportional dimension of finite-dimensional normed spaces: Approach through the isoperimetric inequality. In Banach Spaces (Columbia, Mo., 1984). Lecture Notes in Math. 1166 106–115. Springer, Berlin.
• [16] Milman, V. D. and Schechtman, G. (1986). Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Math. 1200. Springer, Berlin.
• [17] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Univ. Press, Cambridge.
• [18] Pajor, A. and Tomczak-Jaegermann, N. (1986). Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Amer. Math. Soc. 97 637–642.
• [19] Pisier, G. (1989). The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics 94. Cambridge Univ. Press, Cambridge.
• [20] Roberts, P. H. and Ursell, H. D. (1960). Random walk on a sphere and on a Riemannian manifold. Philos. Trans. Roy. Soc. London. Ser. A 252 317–356.
• [21] Shorack, G. R. and Wellner, J. A. (2009). Empirical Processes with Applications to Statistics. Classics in Applied Mathematics 59. SIAM, Philadelphia, PA.
• [22] Vershynin, R. Estimation in high dimensions: A geometric perspective. Available at arXiv:1405.5103.
• [23] Vershynin, R. (2012). Introduction to the non-asymptotic analysis of random matrices. In Compressed Sensing 210–268. Cambridge Univ. Press, Cambridge.